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/* Flowchart variables */
/* Sequence Diagram variables */
/* Gantt chart variables */
/* state colors */
.label {
  
  color: #333; }

.label text {
  fill: #333; }

.node rect,
.node circle,
.node ellipse,
.node polygon {
  fill: #BDD5EA;
  stroke: #9370DB;
  stroke-width: 1px; }

.node .label {
  text-align: center; }

.node.clickable {
  cursor: pointer; }

.arrowheadPath {
  fill: lightgrey; }

.edgePath .path {
  stroke: lightgrey;
  stroke-width: 1.5px; }

.edgeLabel {
  background-color: #e8e8e8;
  text-align: center; }

.cluster rect {
  fill: #6D6D65;
  stroke: rgba(255, 255, 255, 0.25);
  stroke-width: 1px; }

.cluster text {
  fill: #F9FFFE; }

div.mermaidTooltip {
  position: absolute;
  text-align: center;
  max-width: 200px;
  padding: 2px;
  
  font-size: 12px;
  background: #6D6D65;
  border: 1px solid rgba(255, 255, 255, 0.25);
  border-radius: 2px;
  pointer-events: none;
  z-index: 100; }

.actor {
  stroke: #81B1DB;
  fill: #BDD5EA; }

text.actor {
  fill: black;
  stroke: none; }

.actor-line {
  stroke: lightgrey; }

.messageLine0 {
  stroke-width: 1.5;
  stroke-dasharray: '2 2';
  stroke: lightgrey; }

.messageLine1 {
  stroke-width: 1.5;
  stroke-dasharray: '2 2';
  stroke: lightgrey; }

#arrowhead {
  fill: lightgrey; }

.sequenceNumber {
  fill: white; }

#sequencenumber {
  fill: lightgrey; }

#crosshead path {
  fill: lightgrey !important;
  stroke: lightgrey !important; }

.messageText {
  fill: lightgrey;
  stroke: none; }

.labelBox {
  stroke: #81B1DB;
  fill: #BDD5EA; }

.labelText {
  fill: #323D47;
  stroke: none; }

.loopText {
  fill: lightgrey;
  stroke: none; }

.loopLine {
  stroke-width: 2;
  stroke-dasharray: '2 2';
  stroke: #81B1DB; }

.note {
  stroke: rgba(255, 255, 255, 0.25);
  fill: #fff5ad; }

.noteText {
  fill: black;
  stroke: none;
  
  font-size: 14px; }

.activation0 {
  fill: #f4f4f4;
  stroke: #666; }

.activation1 {
  fill: #f4f4f4;
  stroke: #666; }

.activation2 {
  fill: #f4f4f4;
  stroke: #666; }

/** Section styling */
.section {
  stroke: none;
  opacity: 0.2; }

.section0 {
  fill: rgba(255, 255, 255, 0.3); }

.section2 {
  fill: #EAE8B9; }

.section1,
.section3 {
  fill: white;
  opacity: 0.2; }

.sectionTitle0 {
  fill: #F9FFFE; }

.sectionTitle1 {
  fill: #F9FFFE; }

.sectionTitle2 {
  fill: #F9FFFE; }

.sectionTitle3 {
  fill: #F9FFFE; }

.sectionTitle {
  text-anchor: start;
  font-size: 11px;
  text-height: 14px;
   }

/* Grid and axis */
.grid .tick {
  stroke: lightgrey;
  opacity: 0.3;
  shape-rendering: crispEdges; }

.grid path {
  stroke-width: 0; }

/* Today line */
.today {
  fill: none;
  stroke: #DB5757;
  stroke-width: 2px; }

/* Task styling */
/* Default task */
.task {
  stroke-width: 2; }

.taskText {
  text-anchor: middle;
   }

.taskText:not([font-size]) {
  font-size: 11px; }

.taskTextOutsideRight {
  fill: #323D47;
  text-anchor: start;
  font-size: 11px;
   }

.taskTextOutsideLeft {
  fill: #323D47;
  text-anchor: end;
  font-size: 11px; }

/* Special case clickable */
.task.clickable {
  cursor: pointer; }

.taskText.clickable {
  cursor: pointer;
  fill: #003163 !important;
  font-weight: bold; }

.taskTextOutsideLeft.clickable {
  cursor: pointer;
  fill: #003163 !important;
  font-weight: bold; }

.taskTextOutsideRight.clickable {
  cursor: pointer;
  fill: #003163 !important;
  font-weight: bold; }

/* Specific task settings for the sections*/
.taskText0,
.taskText1,
.taskText2,
.taskText3 {
  fill: #323D47; }

.task0,
.task1,
.task2,
.task3 {
  fill: #BDD5EA;
  stroke: rgba(255, 255, 255, 0.5); }

.taskTextOutside0,
.taskTextOutside2 {
  fill: lightgrey; }

.taskTextOutside1,
.taskTextOutside3 {
  fill: lightgrey; }

/* Active task */
.active0,
.active1,
.active2,
.active3 {
  fill: #81B1DB;
  stroke: rgba(255, 255, 255, 0.5); }

.activeText0,
.activeText1,
.activeText2,
.activeText3 {
  fill: #323D47 !important; }

/* Completed task */
.done0,
.done1,
.done2,
.done3 {
  stroke: grey;
  fill: lightgrey;
  stroke-width: 2; }

.doneText0,
.doneText1,
.doneText2,
.doneText3 {
  fill: #323D47 !important; }

/* Tasks on the critical line */
.crit0,
.crit1,
.crit2,
.crit3 {
  stroke: #E83737;
  fill: #E83737;
  stroke-width: 2; }

.activeCrit0,
.activeCrit1,
.activeCrit2,
.activeCrit3 {
  stroke: #E83737;
  fill: #81B1DB;
  stroke-width: 2; }

.doneCrit0,
.doneCrit1,
.doneCrit2,
.doneCrit3 {
  stroke: #E83737;
  fill: lightgrey;
  stroke-width: 2;
  cursor: pointer;
  shape-rendering: crispEdges; }

.milestone {
  transform: rotate(45deg) scale(0.8, 0.8); }

.milestoneText {
  font-style: italic; }

.doneCritText0,
.doneCritText1,
.doneCritText2,
.doneCritText3 {
  fill: #323D47 !important; }

.activeCritText0,
.activeCritText1,
.activeCritText2,
.activeCritText3 {
  fill: #323D47 !important; }

.titleText {
  text-anchor: middle;
  font-size: 18px;
  fill: #323D47;
   }

g.classGroup text {
  fill: #9370DB;
  stroke: none;
  
  font-size: 10px; }
  g.classGroup text .title {
    font-weight: bolder; }

g.classGroup rect {
  fill: #BDD5EA;
  stroke: #9370DB; }

g.classGroup line {
  stroke: #9370DB;
  stroke-width: 1; }

.classLabel .box {
  stroke: none;
  stroke-width: 0;
  fill: #BDD5EA;
  opacity: 0.5; }

.classLabel .label {
  fill: #9370DB;
  font-size: 10px; }

.relation {
  stroke: #9370DB;
  stroke-width: 1;
  fill: none; }

#compositionStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#compositionEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#aggregationStart {
  fill: #BDD5EA;
  stroke: #9370DB;
  stroke-width: 1; }

#aggregationEnd {
  fill: #BDD5EA;
  stroke: #9370DB;
  stroke-width: 1; }

#dependencyStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#dependencyEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#extensionStart {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

#extensionEnd {
  fill: #9370DB;
  stroke: #9370DB;
  stroke-width: 1; }

.commit-id,
.commit-msg,
.branch-label {
  fill: lightgrey;
  color: lightgrey;
   }

.pieTitleText {
  text-anchor: middle;
  font-size: 25px;
  fill: #eee;
}

g.stateGroup text {
  stroke: none;
  font-size: 10px;
}

g.stateGroup circle {
  fill: white !important;
  stroke: white !important;
}

g.stateGroup .state-title {
  font-weight: bolder;
  fill: black; }

g.stateGroup rect {
  fill: #ececff;
  stroke: #9370DB; }

g.stateGroup line {
  stroke: #9370DB;
  stroke-width: 1; }

.transition {
  stroke: #9370DB;
  stroke-width: 1;
  fill: none; }

.stateGroup .composit {
  fill: #555;
  border-bottom: 1px; }

.state-note {
  stroke: rgba(255, 255, 255, 0.25);
  fill: #fff5ad; }
  .state-note text {
    fill: black;
    stroke: none;
    font-size: 10px; }

.stateLabel .box {
  stroke: none;
  stroke-width: 0;
  fill: #BDD5EA;
  opacity: 0.5; }

.stateLabel text {
  fill: black;
  font-size: 10px;
  font-weight: bold;
}

.cluster-label {
  color:black;
}

.statediagram-cluster rect {
  fill: #BDD5EA;
  stroke: #9370DB; 
  stroke-width: 1px;
}
.statediagram-cluster rect.outer {
  rx: 5px;
  ry: 5px;
}
.statediagram-state .divider {
  stroke: #9370DB; 
}

.statediagram-state .title-state {
  rx: 5px;
  ry: 5px;
}
.statediagram-cluster.statediagram-cluster .inner {
  fill: white;
}
.statediagram-cluster.statediagram-cluster-alt .inner {
  fill: #e0e0e0;
}

.statediagram-cluster .inner {
  rx:0;
  ry:0;
}

.statediagram-state rect.basic {
  rx: 5px;
  ry: 5px;
}
.statediagram-state rect.divider {
  stroke-dasharray: 10,10;
  fill: #efefef;
}

.note-edge {
  stroke-dasharray: 5;
}

.statediagram-note rect {
  stroke: var(--cluster-border);
  fill: #fff5ad;
  stroke-width: 1px;
  rx: 0;
  ry: 0;
}

.node circle.state-start {
  fill: black;
  stroke: black;
}
.node circle.state-end {
  fill: black;
  stroke: white;
  stroke-width: 1.5
}
#statediagram-barbEnd {
  fill: #9370DB; 
}

/* CSS Document */

/** code highlight */

.cm-s-inner .cm-variable,
.cm-s-inner .cm-operator,
.cm-s-inner .cm-property {
    color: #b8bfc6;
}

.cm-s-inner .cm-keyword {
    color: #C88FD0;
}

.cm-s-inner .cm-tag {
    color: #7DF46A;
}

.cm-s-inner .cm-attribute {
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}

.CodeMirror div.CodeMirror-cursor {
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}

.cm-s-inner .cm-string {
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}

.cm-s-inner .cm-comment,
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.cm-s-inner .cm-def,
.cm-s-inner.cm-header,
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.cm-s-inner .cm-hr {
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.cm-s-inner .cm-link {
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}

.cm-s-inner .cm-positive {
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}

.cm-s-inner .cm-string-2 {
    color: #f50;
}

.cm-s-inner .cm-meta,
.cm-s-inner .cm-qualifier {
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.cm-s-inner .cm-builtin {
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.cm-s-inner .cm-bracket {
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.cm-s-inner .cm-atom,
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.cm-s-inner .cm-number {
    color: #64AB8F;
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.cm-s-inner .cm-variable {
    color: #b8bfc6;
}

.cm-s-inner .cm-variable-2 {
    color: #9FBAD5;
}

.cm-s-inner .cm-variable-3 {
    color: #1cc685;
}

.CodeMirror-selectedtext,
.CodeMirror-selected {
    background: #4a89dc;
    color: #fff !important;
    text-shadow: none;
}

.CodeMirror-gutters {
    border-right: none;
}

/* CSS Document */

/** markdown source **/
.cm-s-typora-default .cm-header, 
.cm-s-typora-default .cm-property
{
    color: #cebcca;
}

.CodeMirror.cm-s-typora-default div.CodeMirror-cursor{
    border-left: 3px solid #b8bfc6;
}

.cm-s-typora-default .cm-comment {
    color: #9FB1FF;
}

.cm-s-typora-default .cm-string {
    color: #A7A7D9
}

.cm-s-typora-default .cm-atom, .cm-s-typora-default .cm-number {
    color: #848695;
    font-style: italic;
}

.cm-s-typora-default .cm-link {
    color: #95B94B;
}

.cm-s-typora-default .CodeMirror-activeline-background {
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}

.cm-s-typora-default .cm-comment, .cm-s-typora-default .cm-code {
	color: #8aa1e1;
}@import "";
@import "";
@import "";

:root {
    --bg-color:  #363B40;
    --side-bar-bg-color: #2E3033;
    --text-color: #b8bfc6;

    --select-text-bg-color:#4a89dc;

    --item-hover-bg-color: #0a0d16;
    --control-text-color: #b7b7b7;
    --control-text-hover-color: #eee;
    --window-border: 1px solid #555;

    --active-file-bg-color: rgb(34, 34, 34);
    --active-file-border-color: #8d8df0;

    --primary-color: #a3d5fe;

    --active-file-text-color: white;
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    --item-hover-text-color: white;
    --primary-color: #6dc1e7;

    --rawblock-edit-panel-bd: #333;

    --search-select-bg-color: #428bca;
}

html {
    font-size: 16px;
}

html,
body {
    -webkit-text-size-adjust: 100%;
    -ms-text-size-adjust: 100%;
    background: #363B40;
    background: var(--bg-color);
    fill: currentColor;
    line-height: 1.625rem;
}

#write {
    max-width: 1080px;
}


@media only screen and (min-width: 1400px) {
	#write {
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}

@media only screen and (min-width: 1800px) {
	#write {
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}

html,
body,
button,
input,
select,
textarea,
div.code-tooltip-content {
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    border-color: transparent;
}

div.code-tooltip,
.md-hover-tip .md-arrow:after {
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}

.popover.bottom > .arrow:after {
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}

html,
body,
button,
input,
select,
textarea {
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}

hr {
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h2,
h3,
h4,
h5,
h6 {
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}

h1 {
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    line-height: 2.75rem;
    /* 40px */
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}

h2 {
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    /* 30px */
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}

h3 {
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    /* 24px */
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}

h4 {
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}

h5 {
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h6 {
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    h4.md-focus:before,
    h5.md-focus:before,
    h6.md-focus:before {
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        border: 1px solid #ddd;
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        top: .357142857rem;
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        font-weight: normal;
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}

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a:hover {
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a:focus {
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sup.md-footnote {
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}

p {
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p,
ul,
dd,
ol,
hr,
address,
pre,
table,
iframe,
.wp-caption,
.wp-audio-shortcode,
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li > blockquote {
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::-moz-selection {
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*.in-text-selection,
::selection {
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}

ul,
ol {
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}

ol {
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ul ul,
ol ol,
ul ol,
ol ul {
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}

b,
th,
dt,
strong {
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}

i,
em,
dfn,
cite {
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}

blockquote {
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    margin: 0 0 1.875rem 1.875rem;
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    margin-top: 35px;
}

pre,
code,
kbd,
tt,
var {
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code,
tt,
var {
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kbd {
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    font-size: 90%;
    color: #fff;
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pre.md-fences {
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.CodeMirror-gutters {
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.enable-diagrams pre.md-fences[lang="sequence"] .code-tooltip,
.enable-diagrams pre.md-fences[lang="flow"] .code-tooltip,
.enable-diagrams pre.md-fences[lang="mermaid"] .code-tooltip {
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code,
kbd,
tt,
var {
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}

table {
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th,
td {
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}

a {
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hr {
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h1 {
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}

a {
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}

a:hover {
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}

.md-inline-math script {
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}

b,
th,
dt,
strong {
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}

mark {
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}

blockquote {
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table a {
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th,
td {
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}

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}

.md-task-list-item {
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    width: 0.875rem;
    height: 0.875rem;
    vertical-align: middle;
    text-align: center;
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    background-color: #363B40;
    margin-top: -0.4rem;
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.md-task-list-item > input:checked:before,
.md-task-list-item > input[checked]:before {
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    line-height: 0.625rem;
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}

#typora-quick-open {
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}

#typora-quick-open input{
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    border: 0;
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}

.typora-quick-open-item {
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    color: inherit;
}

.typora-quick-open-item.active,
.typora-quick-open-item:hover {
    background-color: #4D8BDB;
    color: white;
}

.typora-quick-open-item:hover {
    background-color: rgba(77, 139, 219, 0.8);
}

.typora-search-spinner > div {
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}

#write pre.md-meta-block {
    border-bottom: 1px dashed #ccc;
    background: transparent;
    padding-bottom: 0.6em;
    line-height: 1.6em;
}

.btn,
.btn .btn-default {
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    color: #b8bfc6;
}

.ty-table-edit {
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}

.popover-title {
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}

.md-image>.md-meta {
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}

.md-expand.md-image>.md-meta {
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#write>h3:before,
#write>h4:before,
#write>h5:before,
#write>h6:before {
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#write>h3.md-focus:before {
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#write>h4.md-focus:before {
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.md-toc-item {
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}

#write div.md-toc-tooltip {
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}

.dropdown-menu .btn:hover,
.dropdown-menu .btn:focus,
.md-toc .btn:hover,
.md-toc .btn:focus {
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    background: black;
}

#toc-dropmenu {
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#toc-dropmenu .divider {
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}

.outline-expander:before {
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}

#typora-sidebar {
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}

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}

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}

.outline-item:hover {
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    background-color: #363B40;
    color: white;
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h1.md-focus .md-attr,
h2.md-focus .md-attr,
h3.md-focus .md-attr,
h4.md-focus .md-attr,
h5.md-focus .md-attr,
h6.md-focus .md-attr,
.md-header-span .md-attr {
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    display: inline;
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.md-comment {
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.md-inline-math svg {
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#math-inline-preview .md-arrow:after {
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.modal-content {
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}

.modal-title {
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}

.modal-content input {
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    color: white;
}

.modal-content .input-group-addon {
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}

.modal-backdrop {
    background-color: rgba(174, 174, 174, 0.7);
}

.modal-content .btn-primary {
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}

.md-table-resize-popover {
    background-color: #333;
}

.form-inline .input-group .input-group-addon {
    color: white;
}

#md-searchpanel {
    border-bottom: 1px dashed grey;
}

/** UI for electron */

.context-menu,
#spell-check-panel,
#footer-word-count-info {
    background-color: #42464A;
}

.context-menu.dropdown-menu .divider,
.dropdown-menu .divider {
    background-color: #777777;
}

footer {
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}

@media (max-width: 1000px) {
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    footer:hover {
        color: inherit;
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}

#file-info-file-path .file-info-field-value:hover {
    background-color: #555;
    color: #dedede;
}

.megamenu-content,
.megamenu-opened header {
    background: var(--bg-color);
}

.megamenu-menu-panel h2,
.megamenu-menu-panel h1,
.long-btn {
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}

.megamenu-menu-panel input[type='text'] {
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    border: 0;
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#recent-file-panel-action-btn {
    background: inherit;
    border: 1px grey solid;
}

.megamenu-menu-panel .dropdown-menu > li > a {
    color: inherit;
    background-color: #2F353A;
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.megamenu-menu-panel table td:nth-child(1) {
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.megamenu-menu-panel tbody tr:hover td:nth-child(1) {
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.modal-footer .btn-default, 
.modal-footer .btn-primary,
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.btn-default:hover, .btn-default:focus, .btn-default.focus, .btn-default:active, .btn-default.active, .open > .dropdown-toggle.btn-default {
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}

.modal-header {
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.modal-footer {
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#w-full.mouse-hover,
#w-pin.mouse-hover {
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#w-unpin {
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#top-titlebar, #top-titlebar * {
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#toggle-sourceview-btn:hover {
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}

/** focus mode */
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}

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}

/*diagrams*/
#write .md-focus.md-fences-with-lineno .md-diagram-panel {
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}

.md-diagram-panel-error {
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}

.active-tab-files #info-panel-tab-file,
.active-tab-files #info-panel-tab-file:hover,
.active-tab-outline #info-panel-tab-outline,
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}

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}

.ty-side-sort-btn.active,
.ty-side-sort-btn:hover,
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#sidebar-files-menu {
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    box-shadow: 4px 4px 20px rgba(0, 0, 0, 0.79);
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}

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}

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}

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}

.file-node-background {
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}

.file-library-node.active>.file-node-content,
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}

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}
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#ty-tooltip {
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}

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}

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}

footer.ty-footer {
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}

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.ty-preferences select {
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    background: #333;
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}

.html-for-mac .file-node-open-state, .html-for-mac .file-node-icon {
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::-webkit-scrollbar-thumb {
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::-webkit-scrollbar-thumb:active {
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#typora-sidebar:hover div.sidebar-content-content::-webkit-scrollbar-thumb:horizontal {
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.nav-group-item:active {
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<div id='write'  class=''><h2><a name="第九章连续动作空间的确定性策略" class="md-header-anchor"></a><span>第九章：连续动作空间的确定性策略</span></h2><p><span>为处理连续动作空间的情况，D. Silver 等在文章《Deterministic Policy Gradient Algorithms》中提出了确定性策略的方法。</span></p><p><span>对于连续动作空间里的确定性策略，</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.514ex" height="2.71ex" viewBox="0 -832.7 4096.2 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E4901-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4901-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E4901-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E4901-MJMAIN-2223" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E4901-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E4901-MJMAIN-3B" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z"></path><path stroke-width="0" id="E4901-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4901-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4901-MJMATHI-3C0" x="0" y="0"></use><use xlink:href="#E4901-MJMAIN-28" x="573" y="0"></use><use xlink:href="#E4901-MJMATHI-61" x="962" y="0"></use><use xlink:href="#E4901-MJMAIN-2223" x="1768" y="0"></use><use xlink:href="#E4901-MJMATHI-73" x="2324" y="0"></use><use xlink:href="#E4901-MJMAIN-3B" x="2793" y="0"></use><use xlink:href="#E4901-MJMATHI-3B8" x="3238" y="0"></use><use xlink:href="#E4901-MJMAIN-29" x="3707" y="0"></use></g></svg></span><script type="math/tex">\pi(a \mid s;\theta)</script><span> 并不是通常意义上的函数，它对策略参数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="1.939ex" viewBox="0 -749.6 469 834.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E4903-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4903-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 的梯度 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.449ex" height="2.71ex" viewBox="0 -832.7 4929.2 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E4899-MJMAIN-2207" d="M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z"></path><path stroke-width="0" id="E4899-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4899-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E4899-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E4899-MJMAIN-2223" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E4899-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E4899-MJMAIN-3B" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z"></path><path stroke-width="0" id="E4899-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4899-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4899-MJMAIN-2207" x="0" y="0"></use><use xlink:href="#E4899-MJMATHI-3C0" x="833" y="0"></use><use xlink:href="#E4899-MJMAIN-28" x="1406" y="0"></use><use xlink:href="#E4899-MJMATHI-61" x="1795" y="0"></use><use xlink:href="#E4899-MJMAIN-2223" x="2601" y="0"></use><use xlink:href="#E4899-MJMATHI-73" x="3157" y="0"></use><use xlink:href="#E4899-MJMAIN-3B" x="3626" y="0"></use><use xlink:href="#E4899-MJMATHI-3B8" x="4071" y="0"></use><use xlink:href="#E4899-MJMAIN-29" x="4540" y="0"></use></g></svg></span><script type="math/tex">\nabla \pi(a \mid s;\theta)</script><span> 也不复存在。第二章曾提到确定性策略可以表示为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.254ex" height="2.71ex" viewBox="0 -832.7 6567.7 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E4902-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4902-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E4902-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E4902-MJMAIN-3B" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z"></path><path stroke-width="0" id="E4902-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4902-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E4902-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E4902-MJMAIN-2208" d="M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z"></path><path stroke-width="0" id="E4902-MJCAL-53" d="M554 512Q536 512 536 522Q536 525 539 539T542 564Q542 588 528 604Q515 616 482 625T410 635Q374 635 349 624T312 594T295 561T290 532Q290 505 303 482T342 442T378 419T409 404Q435 391 451 383T494 357T535 323T562 282T574 231Q574 133 464 56T220 -22Q138 -22 78 21T18 123Q18 184 61 227T156 274Q178 274 178 263Q178 260 177 258Q172 247 164 239T151 227T136 218L127 213L124 202Q118 186 118 163Q120 124 165 86T292 48Q374 48 423 86T473 186V193Q473 267 347 327Q268 364 239 389Q191 431 191 486Q191 547 242 600T356 679T470 705Q472 705 478 705T489 704Q551 704 596 682T642 610Q642 566 621 545Q592 516 554 512Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4902-MJMATHI-3C0" x="0" y="0"></use><use xlink:href="#E4902-MJMAIN-28" x="573" y="0"></use><use xlink:href="#E4902-MJMATHI-73" x="962" y="0"></use><use xlink:href="#E4902-MJMAIN-3B" x="1431" y="0"></use><use xlink:href="#E4902-MJMATHI-3B8" x="1875" y="0"></use><use xlink:href="#E4902-MJMAIN-29" x="2344" y="0"></use><use xlink:href="#E4902-MJMAIN-2C" x="2733" y="0"></use><use xlink:href="#E4902-MJMAIN-28" x="3456" y="0"></use><use xlink:href="#E4902-MJMATHI-73" x="3845" y="0"></use><use xlink:href="#E4902-MJMAIN-2208" x="4591" y="0"></use><use xlink:href="#E4902-MJCAL-53" x="5536" y="0"></use><use xlink:href="#E4902-MJMAIN-29" x="6178" y="0"></use></g></svg></span><script type="math/tex">\pi(s;\theta),\;(s \in \mathcal S)</script><span> ，这种表示可以绕过由于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.514ex" height="2.71ex" viewBox="0 -832.7 4096.2 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E4901-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4901-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E4901-MJMATHI-61" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path><path stroke-width="0" id="E4901-MJMAIN-2223" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E4901-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E4901-MJMAIN-3B" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z"></path><path stroke-width="0" id="E4901-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4901-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4901-MJMATHI-3C0" x="0" y="0"></use><use xlink:href="#E4901-MJMAIN-28" x="573" y="0"></use><use xlink:href="#E4901-MJMATHI-61" x="962" y="0"></use><use xlink:href="#E4901-MJMAIN-2223" x="1768" y="0"></use><use xlink:href="#E4901-MJMATHI-73" x="2324" y="0"></use><use xlink:href="#E4901-MJMAIN-3B" x="2793" y="0"></use><use xlink:href="#E4901-MJMATHI-3B8" x="3238" y="0"></use><use xlink:href="#E4901-MJMAIN-29" x="3707" y="0"></use></g></svg></span><script type="math/tex">\pi(a \mid s;\theta)</script><span> 并不是通常意义上的函数而带来的困难。</span></p><p><span>当策略是一个连续动作空间上的确定性策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.254ex" height="2.71ex" viewBox="0 -832.7 6567.7 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E4902-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4902-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 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type="math/tex">\pi(s;\theta),\;(s \in \mathcal S)</script><span> 时，策略梯度定理为：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n5" cid="n5" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-3316-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="98.296ex" height="7.34ex" viewBox="0 -1829.4 42321.7 3160.5" role="img" focusable="false" style="vertical-align: -3.091ex; max-width: 100%;"><defs><path stroke-width="0" id="E4881-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" 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v_{\pi(\theta)}(s) &= q_{\pi(\theta)}(s, \pi(s;\theta))\;, & s \in \mathcal S \\
q_{\pi(\theta)}(s, \pi(s;\theta)) &= r(s, \pi(s;\theta)) + \gamma\sum_{s'} p(s' \mid s, \pi(s;\theta)) v_{\pi(\theta)}(s')\;, \qquad & s \in \mathcal S
\end{split}</script></div></div><p><span>以上两式对 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="1.939ex" viewBox="0 -749.6 469 834.7" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E4903-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4903-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script><span> 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x="17476" y="0"></use><use xlink:href="#E4883-MJMAIN-28" x="17979" y="0"></use><g transform="translate(18368,0)"><use xlink:href="#E4883-MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4883-MJMAIN-2032" x="663" y="583"></use></g><use xlink:href="#E4883-MJMAIN-2223" x="19410" y="0"></use><use xlink:href="#E4883-MJMATHI-73" x="19966" y="0"></use><use xlink:href="#E4883-MJMAIN-2C" x="20435" y="0"></use><use xlink:href="#E4883-MJMATHI-3C0" x="20879" y="0"></use><use xlink:href="#E4883-MJMAIN-28" x="21452" y="0"></use><use xlink:href="#E4883-MJMATHI-73" x="21841" y="0"></use><use xlink:href="#E4883-MJMAIN-3B" x="22310" y="0"></use><use xlink:href="#E4883-MJMATHI-3B8" x="22755" y="0"></use><use xlink:href="#E4883-MJMAIN-29" x="23224" y="0"></use><use xlink:href="#E4883-MJMAIN-29" x="23613" y="0"></use><use xlink:href="#E4883-MJMAIN-2207" x="24002" y="0"></use><g transform="translate(24835,0)"><use xlink:href="#E4883-MJMATHI-76" x="0" y="0"></use><g transform="translate(485,-186)"><use transform="scale(0.707)" xlink:href="#E4883-MJMATHI-3C0" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4883-MJMAIN-28" x="573" y="0"></use><use transform="scale(0.707)" xlink:href="#E4883-MJMATHI-3B8" x="962" y="0"></use><use transform="scale(0.707)" xlink:href="#E4883-MJMAIN-29" x="1431" y="0"></use></g></g><use xlink:href="#E4883-MJMAIN-28" x="26707" y="0"></use><g transform="translate(27096,0)"><use xlink:href="#E4883-MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4883-MJMAIN-2032" x="663" y="583"></use></g><use xlink:href="#E4883-MJMAIN-29" x="27859" y="0"></use><use xlink:href="#E4883-MJMAIN-2C" x="28526" y="0"></use><use xlink:href="#E4883-MJMATHI-73" x="29971" y="0"></use><use xlink:href="#E4883-MJMAIN-2208" x="30717" y="0"></use><use xlink:href="#E4883-MJCAL-53" x="31662" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3318">\begin{split}
& \nabla v_{\pi(\theta)}(s) = \nabla q_{\pi(\theta)}(s, \pi(s;\theta))\;,\quad s \in \mathcal S \\
& \nabla q_{\pi(\theta)}(s, \pi(s;\theta)) = \big[\nabla_a r(s,a)\big]_{a=\pi(s;\theta)} \nabla\pi(s;\theta) + \\
&\qquad \gamma\sum_{s'} \left\{ \big[\nabla_ap(s' \mid s,a)\big]_{a=\pi(s;\theta)} \big[\nabla\pi(s;\theta)\big] v_{\pi(\theta)}(s') + p(s' \mid s, \pi(s;\theta))\nabla v_{\pi(\theta)}(s') \right\} \\
&= \nabla\pi(s;\theta) \left[\nabla_a r(s,a) + \gamma\sum_{s'} \nabla_ap(s' \mid s,a) v_{\pi(\theta)}(s')\right]_{a=\pi(s;\theta)} + \gamma\sum_{s'}p(s' \mid s, \pi(s;\theta))\nabla v_{\pi(\theta)}(s') \\
&= \nabla\pi(s;\theta) \big[\nabla_a q_{\pi(\theta)}(s, a)\big]_{a=\pi(s;\theta)} + \gamma\sum_{s'}p(s' \mid s, \pi(s;\theta))\nabla v_{\pi(\theta)}(s')\;,\quad s \in \mathcal S
\end{split}</script></div></div><p><span>将 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.47ex" height="2.903ex" viewBox="0 -832.7 7091.3 1250" role="img" focusable="false" style="vertical-align: -0.969ex;"><defs><path stroke-width="0" id="E4904-MJMAIN-2207" d="M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z"></path><path stroke-width="0" id="E4904-MJMATHI-71" d="M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"></path><path stroke-width="0" id="E4904-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4904-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 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405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E4904-MJMAIN-2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path><path stroke-width="0" id="E4904-MJMAIN-3B" d="M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z"></path></defs><g 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xlink:href="#E4904-MJMATHI-3B8" x="5844" y="0"></use><use xlink:href="#E4904-MJMAIN-29" x="6313" y="0"></use><use xlink:href="#E4904-MJMAIN-29" x="6702" y="0"></use></g></svg></span><script type="math/tex">\nabla q_{\pi(\theta)}(s, \pi(s;\theta))</script><span> 的表达式代入 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.179ex" height="2.903ex" viewBox="0 -832.7 3951.9 1250" role="img" focusable="false" style="vertical-align: -0.969ex;"><defs><path stroke-width="0" id="E4906-MJMAIN-2207" d="M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z"></path><path stroke-width="0" id="E4906-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path stroke-width="0" id="E4906-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4906-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E4906-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4906-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E4906-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 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y="0"></use><use transform="scale(0.707)" xlink:href="#E4906-MJMAIN-29" x="1431" y="0"></use></g></g><use xlink:href="#E4906-MJMAIN-28" x="2704" y="0"></use><use xlink:href="#E4906-MJMATHI-73" x="3093" y="0"></use><use xlink:href="#E4906-MJMAIN-29" x="3562" y="0"></use></g></svg></span><script type="math/tex">\nabla v_{\pi(\theta)}(s)</script><span> 的表达式中，再对 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="9.179ex" height="2.903ex" viewBox="0 -832.7 3951.9 1250" role="img" focusable="false" style="vertical-align: -0.969ex;"><defs><path stroke-width="0" id="E4906-MJMAIN-2207" d="M46 676Q46 679 51 683H781Q786 679 786 676Q786 674 617 326T444 -26Q439 -33 416 -33T388 -26Q385 -22 216 326T46 676ZM697 596Q697 597 445 597T193 596Q195 591 319 336T445 80L697 596Z"></path><path stroke-width="0" id="E4906-MJMATHI-76" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 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x="573" y="0"></use><use transform="scale(0.707)" xlink:href="#E4906-MJMATHI-3B8" x="962" y="0"></use><use transform="scale(0.707)" xlink:href="#E4906-MJMAIN-29" x="1431" y="0"></use></g></g><use xlink:href="#E4906-MJMAIN-28" x="2704" y="0"></use><use xlink:href="#E4906-MJMATHI-73" x="3093" y="0"></use><use xlink:href="#E4906-MJMAIN-29" x="3562" y="0"></use></g></svg></span><script type="math/tex">\nabla v_{\pi(\theta)}(s)</script><span> 求关于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.249ex" height="2.228ex" viewBox="0 -749.6 968.3 959.2" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E4917-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 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transform="scale(0.707)" xlink:href="#E4884-MJMAIN-3B" x="3237" y="0"></use><use transform="scale(0.707)" xlink:href="#E4884-MJMATHI-3B8" x="3515" y="0"></use><use transform="scale(0.707)" xlink:href="#E4884-MJMAIN-29" x="3984" y="0"></use></g></g><use xlink:href="#E4884-MJSZ2-5D" x="14424" y="-1"></use></g><use xlink:href="#E4884-MJMAIN-2B" x="17299" y="0"></use><use xlink:href="#E4884-MJMATHI-3B3" x="18299" y="0"></use><use xlink:href="#E4884-MJMAIN-45" x="18842" y="0"></use><use xlink:href="#E4884-MJMAIN-5B" x="19523" y="0"></use><use xlink:href="#E4884-MJMAIN-2207" x="19801" y="0"></use><g transform="translate(20634,0)"><use xlink:href="#E4884-MJMATHI-76" x="0" y="0"></use><g transform="translate(485,-186)"><use transform="scale(0.707)" xlink:href="#E4884-MJMATHI-3C0" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4884-MJMAIN-28" x="573" y="0"></use><use transform="scale(0.707)" xlink:href="#E4884-MJMATHI-3B8" x="962" y="0"></use><use transform="scale(0.707)" xlink:href="#E4884-MJMAIN-29" x="1431" y="0"></use></g></g><use xlink:href="#E4884-MJMAIN-28" x="22506" y="0"></use><g transform="translate(22895,0)"><use xlink:href="#E4884-MJMATHI-53" x="0" y="0"></use><g transform="translate(613,-150)"><use transform="scale(0.707)" xlink:href="#E4884-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4884-MJMAIN-2B" x="361" y="0"></use><use transform="scale(0.707)" xlink:href="#E4884-MJMAIN-31" x="1139" y="0"></use></g></g><use xlink:href="#E4884-MJMAIN-29" x="24767" y="0"></use><use xlink:href="#E4884-MJMAIN-5D" x="25156" y="0"></use></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3319">\begin{split}
& {\rm E}[\nabla v_{\pi(\theta)}(S_t)] = \sum_s {\rm Pr}[S_t=s] \nabla v_{\pi(\theta)}(S_t) \\
&= \sum_s{\rm Pr}[S_t=s] \left[\nabla\pi(s;\theta) \big[\nabla_a q_{\pi(\theta)}(s, a)\big]_{a=\pi(s;\theta)} + \gamma\sum_{s'}p(s' \mid s, \pi(s;\theta))\nabla v_{\pi(\theta)}(s') \right] \\
&= \sum_s{\rm Pr}[S_t=s] \left[\nabla\pi(s;\theta) \big[\nabla_a q_{\pi(\theta)}(s, a)\big]_{a=\pi(s;\theta)} + \gamma\sum_{s'}{\rm Pr}[S_{t+1}=s' \mid S_t=s;\pi(\theta)]\nabla v_{\pi(\theta)}(s') \right] \\
&= \sum_s{\rm Pr}[S_t=s] \nabla\pi(s;\theta) \big[\nabla_a q_{\pi(\theta)}(s, a)\big]_{a=\pi(s;\theta)} + \gamma\sum_s{\rm Pr}[S_t=s]\sum_{s'}{\rm Pr}[S_{t+1}=s' \mid S_t=s;\pi(\theta)]\nabla v_{\pi(\theta)}(s') \\
&= \sum_s{\rm Pr}[S_t=s] \nabla\pi(s;\theta) \big[\nabla_a q_{\pi(\theta)}(s, a)\big]_{a=\pi(s;\theta)} + \gamma\sum_{s'}{\rm Pr}[S_{t+1}=s';\pi(\theta)] \nabla v_{\pi(\theta)}(s') \\
&= {\rm E} \left[\nabla\pi(S_t;\theta)\big[\nabla_a q_{\pi(\theta)}(S_t, a)\big]_{a=\pi(S_t;\theta)} \right] + \gamma{\rm E}[\nabla v_{\pi(\theta)}(S_{t+1})]
\end{split}</script></div></div><p><span>这样就得到了 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.211ex" height="2.903ex" viewBox="0 -832.7 5688.2 1250" role="img" focusable="false" style="vertical-align: -0.969ex;"><defs><path stroke-width="0" id="E4910-MJMAIN-45" d="M128 619Q121 626 117 628T101 631T58 634H25V680H597V676Q599 670 611 560T625 444V440H585V444Q584 447 582 465Q578 500 570 526T553 571T528 601T498 619T457 629T411 633T353 634Q266 634 251 633T233 622Q233 622 233 621Q232 619 232 497V376H286Q359 378 377 385Q413 401 416 469Q416 471 416 473V493H456V213H416V233Q415 268 408 288T383 317T349 328T297 330Q290 330 286 330H232V196V114Q232 57 237 52Q243 47 289 47H340H391Q428 47 452 50T505 62T552 92T584 146Q594 172 599 200T607 247T612 270V273H652V270Q651 267 632 137T610 3V0H25V46H58Q100 47 109 49T128 61V619Z"></path><path stroke-width="0" id="E4910-MJMAIN-5B" d="M118 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284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4910-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E4910-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 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y="0"></use><use xlink:href="#E4910-MJMAIN-2207" x="959" y="0"></use><g transform="translate(1792,0)"><use xlink:href="#E4910-MJMATHI-76" x="0" y="0"></use><g transform="translate(485,-186)"><use transform="scale(0.707)" xlink:href="#E4910-MJMATHI-3C0" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4910-MJMAIN-28" x="573" y="0"></use><use transform="scale(0.707)" xlink:href="#E4910-MJMATHI-3B8" x="962" y="0"></use><use transform="scale(0.707)" xlink:href="#E4910-MJMAIN-29" x="1431" y="0"></use></g></g><use xlink:href="#E4910-MJMAIN-28" x="3663" y="0"></use><g transform="translate(4052,0)"><use xlink:href="#E4910-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4910-MJMATHI-74" x="866" y="-213"></use></g><use xlink:href="#E4910-MJMAIN-29" x="5021" y="0"></use><use xlink:href="#E4910-MJMAIN-5D" x="5410" y="0"></use></g></svg></span><script type="math/tex">{\rm E}[\nabla v_{\pi(\theta)}(S_t)]</script><span> 到 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="15.31ex" height="2.903ex" viewBox="0 -832.7 6591.9 1250" role="img" focusable="false" style="vertical-align: -0.969ex;"><defs><path stroke-width="0" id="E4911-MJMAIN-45" d="M128 619Q121 626 117 628T101 631T58 634H25V680H597V676Q599 670 611 560T625 444V440H585V444Q584 447 582 465Q578 500 570 526T553 571T528 601T498 619T457 629T411 633T353 634Q266 634 251 633T233 622Q233 622 233 621Q232 619 232 497V376H286Q359 378 377 385Q413 401 416 469Q416 471 416 473V493H456V213H416V233Q415 268 408 288T383 317T349 328T297 330Q290 330 286 330H232V196V114Q232 57 237 52Q243 47 289 47H340H391Q428 47 452 50T505 62T552 92T584 146Q594 172 599 200T607 247T612 270V273H652V270Q651 267 632 137T610 3V0H25V46H58Q100 47 109 49T128 61V619Z"></path><path stroke-width="0" id="E4911-MJMAIN-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path stroke-width="0" id="E4911-MJMAIN-2207" d="M46 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transform="translate(1661,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(41.533) matrix(1 0 0 -1 0 0)">边</text></g><g transform="translate(2491,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(41.533) matrix(1 0 0 -1 0 0)">，</text></g><g transform="translate(3322,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(41.533) matrix(1 0 0 -1 0 0)">证</text></g><g transform="translate(4153,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(41.533) matrix(1 0 0 -1 0 0)">毕</text></g><g transform="translate(4983,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(41.533) matrix(1 0 0 -1 0 0)">。</text></g></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3320">\begin{split}
& \nabla {\rm E}_{\pi_{(\theta)}}[G_0] = {\rm E}[\nabla v_{\pi(\theta)}(S_0)] \\
&= {\rm E} \left[\nabla\pi(S_0;\theta)\big[\nabla_a q_{\pi(\theta)}(S_0, a)\big]_{a=\pi(S_0;\theta)} \right] + \gamma{\rm E}[\nabla v_{\pi(\theta)}(S_1)] \\
&= {\rm E} \left[\nabla\pi(S_0;\theta)\big[\nabla_a q_{\pi(\theta)}(S_0, a)\big]_{a=\pi(S_0;\theta)} \right] + \gamma{\rm E} \left[\nabla\pi(S_1;\theta)\big[\nabla_a q_{\pi(\theta)}(S_1, a)\big]_{a=\pi(S_1;\theta)} \right]  + \gamma^2{\rm E}[\nabla v_{\pi(\theta)}(S_2)] \\
&= \cdots = \sum_{t=0}^{+\infty}{\rm E}\left[\gamma^t\nabla\pi(S_t;\theta)\big[\nabla_a q_{\pi(\theta)}(S_t, a)\big]_{a=\pi(S_t;\theta)} \right]=\eqref{eq:1}\text{式右边，证毕。}
\end{split}</script></div></div><p><span>对于连续动作空间中的确定性策略，更常使用的是另外一种形式：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n15" cid="n15" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-3321-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="98.296ex" height="4.447ex" viewBox="0 -1206.4 42321.7 1914.5" role="img" focusable="false" style="vertical-align: -1.645ex; max-width: 100%;"><defs><path stroke-width="0" id="E4886-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" 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xlink:href="#E4887-MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4887-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E4887-MJMAIN-3B" x="20261" y="0"></use><use xlink:href="#E4887-MJMATHI-3B8" x="20706" y="0"></use><use xlink:href="#E4887-MJSZ1-5D" x="21175" y="-1"></use><use xlink:href="#E4887-MJMAIN-64" x="21592" y="0"></use><g transform="translate(22148,0)"><use xlink:href="#E4887-MJMATHI-73" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4887-MJMAIN-30" x="663" y="-213"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3322">\rho_\pi(s) = \int_{s_0 \in \mathcal S} {\large p}_{S_0}(s_0) \sum_{t=0}^{+\infty} \gamma^t {\rm Pr} \big[S_t=s \mid S_0=s_0;\theta \big] {\rm d}s_0</script></div></div><p><span>而言的。证明：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n19" cid="n19" mdtype="math_block"><div 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y="0"></use><use xlink:href="#E4888-MJMATHI-73" x="2267" y="0"></use><use xlink:href="#E4888-MJMAIN-3B" x="2736" y="0"></use><use xlink:href="#E4888-MJMATHI-3B8" x="3180" y="0"></use><use xlink:href="#E4888-MJMAIN-29" x="3649" y="0"></use><g transform="translate(4038,0)"><use xlink:href="#E4888-MJSZ1-5B"></use><g transform="translate(417,0)"><use xlink:href="#E4888-MJMAIN-2207" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMATHI-61" x="1178" y="-213"></use></g><g transform="translate(1724,0)"><use xlink:href="#E4888-MJMATHI-71" x="0" y="0"></use><g transform="translate(446,-186)"><use transform="scale(0.707)" xlink:href="#E4888-MJMATHI-3C0" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMAIN-28" x="573" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMATHI-3B8" x="962" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMAIN-29" x="1431" y="0"></use></g></g><use xlink:href="#E4888-MJMAIN-28" x="3556" y="0"></use><use xlink:href="#E4888-MJMATHI-73" x="3945" y="0"></use><use xlink:href="#E4888-MJMAIN-2C" x="4414" y="0"></use><use xlink:href="#E4888-MJMATHI-61" x="4859" y="0"></use><use xlink:href="#E4888-MJMAIN-29" x="5388" y="0"></use><use xlink:href="#E4888-MJSZ1-5D" x="5777" y="-1"></use><g transform="translate(6194,-399)"><use transform="scale(0.707)" xlink:href="#E4888-MJMATHI-61" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMAIN-3D" x="529" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMATHI-3C0" x="1307" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMAIN-28" x="1880" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMATHI-73" x="2269" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMAIN-3B" x="2738" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMATHI-3B8" x="3016" y="0"></use><use transform="scale(0.707)" xlink:href="#E4888-MJMAIN-29" x="3485" y="0"></use></g></g><use xlink:href="#E4888-MJSZ2-5D" x="13072" y="-1"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3323">\begin{split}
& \nabla {\rm E}_{\pi_{(\theta)}}[G_0] = \sum_{t=0}^{+\infty}{\rm E}\left[\gamma^t\nabla\pi(S_t;\theta)\big[\nabla_a q_{\pi(\theta)}(S_t, a)\big]_{a=\pi(S_t;\theta)} \right] \\
&= \sum_{t=0}^{+\infty} \int_s {\large p}_{S_t}(s) \gamma^t \nabla\pi(s;\theta) \left[\nabla_a q_{\pi(\theta)}(s,a)\right]_{a=\pi({\color \red S_t};\theta)} {\rm d}s \\
&= \sum_{t=0}^{+\infty} \int_s \left(\int_{s_0}{\large p}_{S_0}(s_0) {\rm Pr}[S_t=s \mid S_0=s_0;\theta] {\rm d}s_0 \right) \gamma^t \nabla\pi(s;\theta) \left[\nabla_a q_{\pi(\theta)}(s,a)\right]_{a=\pi(s;\theta)} {\rm d}s \\
&= \int_s \left(\int_{s_0}{\large p}_{S_0}(s_0) \sum_{t=0}^{+\infty} \gamma^t {\rm Pr}[S_t=s \mid S_0=s_0;\theta] {\rm d}s_0 \right) \nabla\pi({\color \red S_t};\theta) \left[\nabla_a q_{\pi(\theta)}(s,a)\right]_{a=\pi(s;\theta)} {\rm d}s \\
&= \int_s \rho_{\pi(\theta)}(s) \nabla\pi(s;\theta) \left[\nabla_a q_{\pi(\theta)}(s,a)\right]_{a=\pi(s;\theta)} {\rm d}s \\
&= {\rm E}_{\large \rho_{\pi(\theta)}} \left[\nabla\pi(s;\theta) \left[\nabla_a q_{\pi(\theta)}(s,a)\right]_{a=\pi(s;\theta)} \right]
\end{split}</script></div></div><p><em><span>(在上式证明中，个人认为标红的项应该改为被积变量 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="1.36ex" viewBox="0 -500.4 469 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E4914-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4914-MJMATHI-73" x="0" y="0"></use></g></svg></span><script type="math/tex">s</script><span> 更为恰当；因为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="1.36ex" viewBox="0 -500.4 469 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E4914-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4914-MJMATHI-73" x="0" y="0"></use></g></svg></span><script type="math/tex">s</script><span> 是随机变量 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.249ex" height="2.228ex" viewBox="0 -749.6 968.3 959.2" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E4917-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 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xlink:href="#E4917-MJMATHI-74" x="866" y="-213"></use></g></svg></span><script type="math/tex">S_t</script><span> 的可能取值，在对 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.249ex" height="2.228ex" viewBox="0 -749.6 968.3 959.2" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E4917-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z"></path><path stroke-width="0" id="E4917-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4917-MJMATHI-53" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4917-MJMATHI-74" x="866" y="-213"></use></g></svg></span><script type="math/tex">S_t</script><span> 求期望的积分表达式中，不应该出现随机变量 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.249ex" height="2.228ex" viewBox="0 -749.6 968.3 959.2" role="img" focusable="false" style="vertical-align: -0.487ex;"><defs><path stroke-width="0" id="E4917-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 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class="md-header-anchor"></a><span>一、同策确定性算法</span></h3><p><span>根据策略梯度定理的确定性版本，将算法 8-1 中策略改进的迭代式和其他与策略相关的部分做相应修改，即可有连续动作空间中的确定性执行者 / 评论者算法。由于确定性策略输出确定，无法对环境进行充分探索，因此在训练的时候需要在策略的动作空间添加扰动实现探索，具体算法如下：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n22" cid="n22" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-3324-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; zoom: 0.971345;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="101.294ex" height="71.488ex" viewBox="-18.1 -43.5 43612.5 30779.6" role="img" focusable="false" style="vertical-align: -71.387ex; margin-left: -0.042ex; max-width: 100%;"><defs><path stroke-width="0" id="E4889-MJMAINB-39" d="M178 59Q206 48 238 48Q311 48 345 102Q370 138 375 259V278Q374 278 369 271T350 252T322 232Q297 220 258 220Q172 220 110 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\large \textbf{算法 9-1   基本的同策确定性执行者/评论者算法} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：环境（无数学描述）。} \\
&\text{输出：最优策略的估计 $\pi(\theta)$ 。} \\
&\text{参数：优化器（隐含学习率 $\alpha^{(\bold w)}, \alpha^{(\theta)}$ ），折扣因子 $\gamma$ ，控制回合数和回合内步数的参数。} \\
&\text{1.（初始化）$\theta \leftarrow$ 任意值，$\bold w \leftarrow$ 任意值。} \\
&\text{2.（带自益的策略更新）对于每个回合执行以下操作：} \\
&\qquad \text{2.1（初始化累计折扣）$I \leftarrow 1$ 。} \\
&\qquad \text{2.2（决定初始状态动作对）选择状态 $S$ ，对 $\pi(S; \theta)$ 加扰动（如正态分布随机变量扰动）得到动作 $A$ 。} \\
&\qquad \text{2.3 $\;\,$如果回合未结束，执行以下操作：} \\
&\qquad \qquad \text{2.3.1（采样）根据状态 $S$ 和动作 $A$ 得到奖励 $R$ 和下一个状态 $S'$ ；} \\
&\qquad \qquad \text{2.3.2（执行）用 $\pi(S'; \theta)$ 加扰动得到 $A'$ ；} \\
&\qquad \qquad \text{2.3.3（估计回报）$U \leftarrow R + \gamma q(S',A';\bold w)$ ；} \\
&\qquad \qquad \text{2.3.4（更新价值）更新 $\bold w$ 以减小 $[U - q(S,A;\bold w)]^2$ ，} \\
&\qquad \qquad \qquad \;\, \text{如 $\bold w \leftarrow \bold w + \alpha^{(\bold w)} [U - q(S,A;\bold w)] \nabla q(S,A;\bold w)$ ；} \\
&\qquad \qquad \text{2.3.5（策略改进）更新 $\theta$ 以减小 $-Iq(S,\pi(S;\theta);\bold w)$ ，} \\
&\qquad \qquad \qquad \;\, \text{如 $\theta \leftarrow \theta + \alpha^{(\theta)} I\nabla\pi(S;\theta)[\nabla_a q(S,a;\bold w)]_{a=\pi(S;\theta)}$ ；} \\
&\qquad \qquad \text{2.3.6（更新累计折扣）$I \leftarrow \gamma I$ ；} \\
&\qquad \qquad \text{2.3.7（更新状态）$S \leftarrow S', A \leftarrow A'$ 。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>在有些任务中，动作的效果经过低通滤波器处理后反映在系统中，而独立同分布的 Gaussian 噪声不能有效实现探索。例如在某个任务中，动作的直接效果是改变一个质点的加速度，如果在这个任务中用独立同分布的 Gaussian 噪声叠加在动作上，那么对质点位置的整体效果是在没有噪声的位置附近移动。这样的探索就没有办法为质点的位置提供持续的偏移，使得质点到比较远的位置。这类任务中，常常用 Ornstein Uhlenbeck 过程作为动作噪声，它是用下列随机微分方程定义的（以一维的情况为例）：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n24" cid="n24" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-3325-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="98.296ex" height="2.71ex" viewBox="0 -832.7 42321.7 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex; max-width: 100%;"><defs><path stroke-width="0" id="E4890-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 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y="-213"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3325">{\rm d}N_t = \theta(\mu - N_t){\rm d}t + \sigma {\rm d}B_t</script></div></div><p><span>其中 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.884ex" height="2.421ex" viewBox="0 -749.6 2533.3 1042.3" role="img" focusable="false" style="vertical-align: -0.68ex;"><defs><path stroke-width="0" id="E4918-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4918-MJMAIN-2C" d="M78 35T78 60T94 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104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4918-MJMATHI-3B8" x="0" y="0"></use><use xlink:href="#E4918-MJMAIN-2C" x="469" y="0"></use><use xlink:href="#E4918-MJMATHI-3BC" x="913" y="0"></use><use xlink:href="#E4918-MJMAIN-2C" x="1516" y="0"></use><use xlink:href="#E4918-MJMATHI-3C3" x="1961" y="0"></use></g></svg></span><script type="math/tex">\theta,\mu,\sigma</script><span> 是参数（ </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="11.968ex" height="2.324ex" viewBox="0 -749.6 5152.8 1000.8" role="img" focusable="false" style="vertical-align: -0.583ex;"><defs><path stroke-width="0" id="E4919-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4919-MJMAIN-3E" d="M84 520Q84 528 88 533T96 539L99 540Q106 540 253 471T544 334L687 265Q694 260 694 250T687 235Q685 233 395 96L107 -40H101Q83 -38 83 -20Q83 -19 83 -17Q82 -10 98 -1Q117 9 248 71Q326 108 378 132L626 250L378 368Q90 504 86 509Q84 513 84 520Z"></path><path stroke-width="0" id="E4919-MJMAIN-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 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xlink:href="#E4891-MJMAIN-29" x="2514" y="0"></use></g></g><use xlink:href="#E4891-MJMAIN-64" x="7431" y="0"></use><g transform="translate(7987,0)"><use xlink:href="#E4891-MJMATHI-42" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4891-MJMATHI-74" x="1073" y="-213"></use></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3326">N_t = \sigma \int_0^t e^{\theta(\tau-t)} {\rm d}B_t</script></div></div><p><span>（证明：略）</span></p><p><span>这个解的均值为 0 ，方差为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.803ex" height="5.411ex" viewBox="0 -1580.2 5942.9 2329.8" role="img" focusable="false" style="vertical-align: -1.741ex;"><defs><path stroke-width="0" id="E4923-MJMATHI-3C3" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path stroke-width="0" id="E4923-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path stroke-width="0" id="E4923-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4923-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E4923-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E4923-MJMAIN-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path stroke-width="0" id="E4923-MJMATHI-65" d="M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"></path><path stroke-width="0" id="E4923-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E4923-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="1145" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#E4923-MJMATHI-3C3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4923-MJMAIN-32" x="809" y="513"></use></g><g transform="translate(88,-686)"><use xlink:href="#E4923-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E4923-MJMATHI-3B8" x="500" y="0"></use></g></g><use xlink:href="#E4923-MJMAIN-28" x="1385" y="0"></use><use xlink:href="#E4923-MJMAIN-31" x="1774" y="0"></use><use xlink:href="#E4923-MJMAIN-2212" x="2497" y="0"></use><g transform="translate(3497,0)"><use xlink:href="#E4923-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#E4923-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4923-MJMAIN-32" x="778" y="0"></use><use transform="scale(0.707)" xlink:href="#E4923-MJMATHI-3B8" x="1278" y="0"></use><use transform="scale(0.707)" xlink:href="#E4923-MJMATHI-74" x="1747" y="0"></use></g></g><use xlink:href="#E4923-MJMAIN-29" x="5553" y="0"></use></g></svg></span><script type="math/tex">\displaystyle \frac{\sigma^2}{2\theta}(1-e^{-2\theta t})</script><span> ，协方差为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="37.622ex" height="5.411ex" viewBox="0 -1580.2 16198.3 2329.8" role="img" focusable="false" style="vertical-align: -1.741ex;"><defs><path stroke-width="0" id="E4924-MJMAIN-43" d="M56 342Q56 428 89 500T174 615T283 681T391 705Q394 705 400 705T408 704Q499 704 569 636L582 624L612 663Q639 700 643 704Q644 704 647 704T653 705H657Q660 705 666 699V419L660 413H626Q620 419 619 430Q610 512 571 572T476 651Q457 658 426 658Q322 658 252 588Q173 509 173 342Q173 221 211 151Q232 111 263 84T328 45T384 29T428 24Q517 24 571 93T626 244Q626 251 632 257H660L666 251V236Q661 133 590 56T403 -21Q262 -21 159 83T56 342Z"></path><path stroke-width="0" id="E4924-MJMAIN-6F" d="M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z"></path><path stroke-width="0" id="E4924-MJMAIN-76" d="M338 431Q344 429 422 429Q479 429 503 431H508V385H497Q439 381 423 345Q421 341 356 172T288 -2Q283 -11 263 -11Q244 -11 239 -2Q99 359 98 364Q93 378 82 381T43 385H19V431H25L33 430Q41 430 53 430T79 430T104 429T122 428Q217 428 232 431H240V385H226Q187 384 184 370Q184 366 235 234L286 102L377 341V349Q377 363 367 372T349 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379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path><path stroke-width="0" id="E4924-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path><path stroke-width="0" id="E4924-MJMAIN-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path stroke-width="0" id="E4924-MJMATHI-3C3" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 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y="-213"></use></g><use xlink:href="#E4924-MJMAIN-29" x="4976" y="0"></use><use xlink:href="#E4924-MJMAIN-3D" x="5643" y="0"></use><g transform="translate(6421,0)"><g transform="translate(397,0)"><rect stroke="none" width="1145" height="60" x="0" y="220"></rect><g transform="translate(60,676)"><use xlink:href="#E4924-MJMATHI-3C3" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-32" x="809" y="513"></use></g><g transform="translate(88,-686)"><use xlink:href="#E4924-MJMAIN-32" x="0" y="0"></use><use xlink:href="#E4924-MJMATHI-3B8" x="500" y="0"></use></g></g></g><use xlink:href="#E4924-MJMAIN-28" x="8085" y="0"></use><g transform="translate(8474,0)"><use xlink:href="#E4924-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMATHI-3B8" x="778" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-7C" x="1246" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMATHI-74" x="1524" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-2212" x="1886" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMATHI-73" x="2664" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-7C" x="3133" y="0"></use></g></g><use xlink:href="#E4924-MJMAIN-2212" x="11674" y="0"></use><g transform="translate(12674,0)"><use xlink:href="#E4924-MJMATHI-65" x="0" y="0"></use><g transform="translate(466,412)"><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-2212" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMATHI-3B8" x="778" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-28" x="1246" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMATHI-74" x="1636" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-2B" x="1997" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMATHI-73" x="2775" y="0"></use><use transform="scale(0.707)" xlink:href="#E4924-MJMAIN-29" x="3244" y="0"></use></g></g><use xlink:href="#E4924-MJMAIN-29" x="15809" y="0"></use></g></svg></span><script type="math/tex">\displaystyle {\rm Cov}(N_t,N_s)=\frac{\sigma^2}{2\theta}(e^{-\theta|t-s|}-e^{-\theta(t+s)})</script><span> ，（证明：略）。</span></p><p><span>对于 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.025ex" height="2.517ex" viewBox="0 -791.1 2163.6 1083.8" role="img" focusable="false" style="vertical-align: -0.68ex;"><defs><path stroke-width="0" id="E4925-MJMATHI-74" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E4925-MJMAIN-2260" d="M166 -215T159 -215T147 -212T141 -204T139 -197Q139 -190 144 -183L306 133H70Q56 140 56 153Q56 168 72 173H327L406 327H72Q56 332 56 347Q56 360 70 367H426Q597 702 602 707Q605 716 618 716Q625 716 630 712T636 703T638 696Q638 692 471 367H707Q722 359 722 347Q722 336 708 328L451 327L371 173H708Q722 163 722 153Q722 140 707 133H351Q175 -210 170 -212Q166 -215 159 -215Z"></path><path stroke-width="0" id="E4925-MJMATHI-73" d="M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4925-MJMATHI-74" x="0" y="0"></use><use xlink:href="#E4925-MJMAIN-2260" x="638" y="0"></use><use xlink:href="#E4925-MJMATHI-73" x="1694" y="0"></use></g></svg></span><script type="math/tex">t \ne s</script><span> 总有 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="13.923ex" height="2.614ex" viewBox="0 -832.7 5994.4 1125.4" role="img" focusable="false" style="vertical-align: -0.68ex;"><defs><path stroke-width="0" id="E4926-MJMAIN-7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path><path stroke-width="0" id="E4926-MJMATHI-74" d="M26 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y="0"></use><use xlink:href="#E4926-MJMAIN-7C" x="2330" y="0"></use><use xlink:href="#E4926-MJMAIN-3C" x="2886" y="0"></use><use xlink:href="#E4926-MJMATHI-74" x="3942" y="0"></use><use xlink:href="#E4926-MJMAIN-2B" x="4525" y="0"></use><use xlink:href="#E4926-MJMATHI-73" x="5525" y="0"></use></g></svg></span><script type="math/tex">|t-s|<t+s</script><span> ，所以 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="16.721ex" height="2.71ex" viewBox="0 -832.7 7199.1 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E4927-MJMAIN-43" d="M56 342Q56 428 89 500T174 615T283 681T391 705Q394 705 400 705T408 704Q499 704 569 636L582 624L612 663Q639 700 643 704Q644 704 647 704T653 705H657Q660 705 666 699V419L660 413H626Q620 419 619 430Q610 512 571 572T476 651Q457 658 426 658Q322 658 252 588Q173 509 173 342Q173 221 211 151Q232 111 263 84T328 45T384 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y="0"></use><use xlink:href="#E4927-MJMAIN-30" x="6699" y="0"></use></g></svg></span><script type="math/tex">{\rm Cov}(N_t,N_s)>0</script><span> ，据此可知，使用 Ornstein Uhlenbeck 过程让相邻扰动正相关，进而让动作向相近的方向偏移。</span></p><p><em><span>(以上 Ornstein Uhlenbeck 过程的内容书中描述很少， 不好理解本质，所以证明省略了，后续需要再查阅相关资料。)</span></em></p><h3><a name="二异策确定性算法" class="md-header-anchor"></a><span>二、异策确定性算法</span></h3><p><span>确定性执行者 / 评论者算法也可以利用行为策略得到其异策版本。但对于确定性算法，行为策略并不对目标策略绝对连续，不再简单使用重采样。考虑非确定性版本的异策执行者 / 评论者算法的策略梯度可表示为：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n68" cid="n68" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-3327-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="98.296ex" height="7.34ex" viewBox="0 -1829.4 42321.7 3160.5" 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xlink:href="#E4931-MJMAIN-29" x="1431" y="0"></use></g></g><use xlink:href="#E4931-MJMAIN-28" x="3672" y="0"></use><use xlink:href="#E4931-MJMATHI-53" x="4061" y="0"></use><use xlink:href="#E4931-MJMAIN-2C" x="4706" y="0"></use><use xlink:href="#E4931-MJMATHI-41" x="5151" y="0"></use><use xlink:href="#E4931-MJMAIN-29" x="5901" y="0"></use><use xlink:href="#E4931-MJMATHI-3C0" x="6290" y="0"></use><use xlink:href="#E4931-MJMAIN-28" x="6863" y="0"></use><use xlink:href="#E4931-MJMATHI-41" x="7252" y="0"></use><use xlink:href="#E4931-MJMAIN-2223" x="8280" y="0"></use><use xlink:href="#E4931-MJMATHI-53" x="8836" y="0"></use><use xlink:href="#E4931-MJMAIN-3B" x="9481" y="0"></use><use xlink:href="#E4931-MJMATHI-3B8" x="9925" y="0"></use><use xlink:href="#E4931-MJMAIN-29" x="10394" y="0"></use><use xlink:href="#E4931-MJMAIN-5D" x="10783" y="0"></use></g></svg></span><script type="math/tex">{\rm E}_{\large \rho_b}[q_{\pi(\theta)}(S,A) \pi(A \mid S;\theta)]</script><span> 实现迭代。确定性版本的异策执行者 / 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\right]</script></div></div><p><span>这个表达式与同策的情形相比，期望运算针对的表达式相同，但是期望针对的分布不同，即行为策略确定了轨迹的分布，它体现在了期望上。由于行为策略能够促进探索，所以异策算法有时会比同策算法性能好。</span></p><p><span>基于以上，可以得到</span><strong><span>异策确定性执行者 / 评论者算法</span></strong><span>（Off-Policy Deterministic Actor-Critic, OPDAC）。由于在异策算法迭代更新策略参数时，对环境使用的是行为策略决定的动作，所以需要额外计算目标策略的动作；在更新价值函数时，采用的是 Q 学习，依然需要计算目标策略的动作。具体算法如下：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n28" cid="n28" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display"><span class="MathJax_SVG" id="MathJax-Element-3329-Frame" tabindex="-1" style="font-size: 100%; display: inline-block; zoom: 0.971345;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="101.294ex" height="42.067ex" viewBox="-18.1 -43.5 43612.5 18112.2" role="img" focusable="false" style="vertical-align: -41.966ex; margin-left: -0.042ex; max-width: 100%;"><defs><path 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\large \textbf{算法 9-2   基本的异策确定性执行者/评论者算法} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\cdots \quad \text{同算法 9-1} \quad \cdots \\
&\text{2.2（初始状态）选择状态 $S$ 。} \\
&\text{2.3 $\;\,$如果回合未结束，执行以下操作：} \\
&\qquad \text{2.3.1（执行）用 $b(\cdot \mid S)$ 得到 $A$ ；} \\
&\qquad \text{2.3.2（采样）根据状态 $S$ 和动作 $A$ 得到奖励 $R$ 和下一个状态 $S'$ ；} \\
&\qquad \text{2.3.3（估计回报）$U \leftarrow R + \gamma q(S',\pi(S';\theta);\bold w)$ ；} \\
&\cdots \quad \text{同算法 9-1} \quad \cdots \\
&\qquad \text{2.3.7（更新状态）$S \leftarrow S'$ 。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>将基本的异策确定性执行者 / 评论者算法和深度 Q 网络中常用的技术（经验回放、目标网络）结合，即可得到</span><strong><span>深度确定性策略梯度算法</span></strong><span>（Deep Deterministic Policy Gradient, DDPG），算法 9-3 给出了该算法。另外，在更新目标网络时，为了避免参数更新过快，还引入了目标网络的学习率 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="12.449ex" height="2.71ex" viewBox="0 -832.7 5359.9 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E4933-MJMATHI-3B1" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 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\large \textbf{算法 9-3   深度确定性策略梯度算法（假设 $\pi(S;\theta)+N$ 总是在动作空间内）} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\text{输入：环境（无数学描述）。} \\
&\text{输出：最优策略的估计 $\pi(\theta)$ 。} \\
&\text{参数：学习率 $\alpha^{(\bold w)}, \alpha^{(\theta)}$ ，折扣因子 $\gamma$ ，控制回合数和回合内步数的参数，目标网络学习率 $\alpha_{目标}$ 。} \\
&\text{1.（初始化）$\theta \leftarrow$ 任意值，$\theta_{目标} \leftarrow \theta$ ，$\bold w \leftarrow$ 任意值，$\bold w_{目标} \leftarrow \bold w$ 。} \\
&\text{2. $\;\,$循环执行以下操作：} \\
&\qquad \text{2.1（积累经验）从起始状态 $S$ 出发，执行以下操作，直到满足终止条件：} \\
&\qquad \qquad \text{2.1.1 $\;\,$对 $\pi(S;\theta)$ 加扰动得到动作 $A$（如用正态分布随机变量扰动）；} \\
&\qquad \qquad \text{2.1.2 $\;\,$执行动作 $A$ ，观测到收益 $R$ 和下一状态 $S'$ ；} \\
&\qquad \qquad \text{2.1.3 $\;\,$将经验 $(S,A,R,S')$ 存储在经验存储空间 $\mathcal D$ 。} \\
&\qquad \text{2.2（更新）在更新的时机，执行一次或多次以下更新操作：} \\
&\qquad \qquad \text{2.2.1（回放）从存储空间 $\mathcal D$ 采样一批经验 $\mathcal B$ ；} \\
&\qquad \qquad \text{2.2.2（估计回报）$U \leftarrow R + \gamma q(S',\pi(S';\theta_{目标});\bold w_{目标})$ ，$(S,A,R,S')\in \mathcal B$ ；} \\
&\qquad \qquad \text{2.2.3（更新价值）更新 $\bold w$ 以减小 $\frac{1}{|\mathcal B|} \sum_{(S,A,R,S')\in \mathcal B} [U - q(S,A;\bold w)]^2$ ；} \\
&\qquad \qquad \text{2.2.4（策略改进）更新 $\theta$ 以减小 $-\frac{1}{|\mathcal B|} \sum_{(S,A,R,S')\in \mathcal B} q(S,\pi(S;\theta);\bold w)$ ，} \\
&\qquad \qquad \qquad \;\, \text{如 $\theta \leftarrow \theta + \alpha^{(\theta)} \frac{1}{|\mathcal B|} \sum_{(S,A,R,S')\in \mathcal B} \nabla\pi(S;\theta)[\nabla_a q(S,a;\bold w)]_{a=\pi(S;\theta)}$ ；} \\
&\qquad \qquad \text{2.2.5（更新目标）在恰当的时机更新目标网络和目标策略，} \\
&\qquad \qquad \qquad \;\, \text{$\bold w_{目标} \leftarrow (1-\alpha_{目标}) \bold w_{目标} + \alpha_{目标}\bold w$ ，$\theta_{目标} \leftarrow (1-\alpha_{目标}) \theta_{目标} + \alpha_{目标}\theta$ 。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script></div></div><p><span>S. Fujimoto 等人在文章《Addressing function approximation error in actor-critic methods》中给出了</span><strong><span>双重延迟深度确定性策略梯度算法</span></strong><span>（Twin Delay Deep Deterministic Policy Gradient, TD3），结合了深度确定性策略梯度算法和双重 Q 学习。</span></p><p><span>对于确定性策略梯度算法，动作已经由含参策略 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.227ex" height="2.71ex" viewBox="0 -832.7 1820 1166.9" role="img" focusable="false" style="vertical-align: -0.776ex;"><defs><path stroke-width="0" id="E4934-MJMATHI-3C0" d="M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z"></path><path stroke-width="0" id="E4934-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path stroke-width="0" id="E4934-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path><path stroke-width="0" id="E4934-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4934-MJMATHI-3C0" x="0" y="0"></use><use xlink:href="#E4934-MJMAIN-28" x="573" y="0"></use><use xlink:href="#E4934-MJMATHI-3B8" x="962" y="0"></use><use xlink:href="#E4934-MJMAIN-29" x="1431" y="0"></use></g></svg></span><script type="math/tex">\pi(\theta)</script><span> 决定了，双重网络则要由双重延迟深度确定性策略梯度算法维护两份学习过程的价值网络参数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.007ex" height="2.421ex" viewBox="0 -957.2 1725.1 1042.3" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E4935-MJMAINB-77" d="M624 444Q636 441 722 441Q797 441 800 444H805V382H741L593 11Q592 10 590 8T586 4T584 2T581 0T579 -2T575 -3T571 -3T567 -4T561 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293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z"></path><path stroke-width="0" id="E4935-MJMAIN-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4935-MJMAINB-77" x="0" y="0"></use><g transform="translate(831,362)"><use transform="scale(0.707)" xlink:href="#E4935-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4935-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E4935-MJMAIN-29" x="733" y="0"></use></g></g></svg></span><script type="math/tex">\bold w^{(i)}</script><span> 和目标网络参数 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="4.891ex" height="3.771ex" viewBox="0 -1123.4 2105.7 1623.8" role="img" focusable="false" style="vertical-align: -1.162ex;"><defs><path stroke-width="0" id="E4936-MJMAINB-77" d="M624 444Q636 441 722 441Q797 441 800 444H805V382H741L593 11Q592 10 590 8T586 4T584 2T581 0T579 -2T575 -3T571 -3T567 -4T561 -4T553 -4H542Q525 -4 518 6T490 70Q474 110 463 137L415 257L367 137Q357 111 341 72Q320 17 313 7T289 -4H277Q259 -4 253 -2T238 11L90 382H25V444H32Q47 441 140 441Q243 441 261 444H270V382H222L310 164L382 342L366 382H303V444H310Q322 441 407 441Q508 441 523 444H531V382H506Q481 382 481 380Q482 376 529 259T577 142L674 382H617V444H624Z"></path><path stroke-width="0" id="E4936-MJMAIN-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 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749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E4936-MJMAINB-77" x="0" y="0"></use><g transform="translate(831,521)"><use transform="scale(0.707)" xlink:href="#E4936-MJMAIN-28" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E4936-MJMATHI-69" x="389" y="0"></use><use transform="scale(0.707)" xlink:href="#E4936-MJMAIN-29" x="733" y="0"></use></g><g transform="translate(831,-326)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(29.368) matrix(1 0 0 -1 0 0)">目</text><g transform="translate(587,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(29.368) matrix(1 0 0 -1 0 0)">标</text></g></g></g></svg></span><script type="math/tex">\bold w_{目标}^{(i)}</script><span> （</span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" 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font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(29.368) matrix(1 0 0 -1 0 0)">目</text><g transform="translate(587,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(29.368) matrix(1 0 0 -1 0 0)">标</text></g></g></g><use xlink:href="#E4938-MJSZ2-29" x="4148" y="-1"></use></g></g></svg></span><script type="math/tex">\displaystyle \min_{i=0,1}q\left(\cdot, \cdot;\bold w_{目标}^{(i)} \right)</script><span> ，具体算法如下：</span></p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n88" cid="n88" mdtype="math_block"><div class="md-rawblock-container md-math-container" contenteditable="false" tabindex="-1">
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0 0)">标</text></g></g></g><use xlink:href="#E4947-MJMATHI-3B8" x="12595" y="0"></use></g><g transform="translate(23347,0)"><g transform="translate(250,0)"><text font-family="STIXGeneral, 'PingFang SC', serif" stroke="none" transform="scale(41.533) matrix(1 0 0 -1 0 0)">。</text></g></g></g></g><g transform="translate(0,-10899)"><g><rect fill="black" stroke="none" width="41597" height="100" x="0" y="-500"></rect></g></g></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-3381">\; \\ \; \\
\large \textbf{算法 9-4   双重延迟深度确定性策略梯度算法} \\
\begin{split}
\rule[5pt]{10mm}{0.1em} &\rule[5pt]{265mm}{0.1em} \\
&\cdots \quad \text{同算法 9-3} \quad \cdots \\
&\text{1.（初始化）$\theta \leftarrow$ 任意值，$\theta_{目标} \leftarrow \theta$ ，$\bold w^{(i)} \leftarrow$ 任意值，$\bold w_{目标}^{(i)} \leftarrow \bold w^{(i)}$ ，$i \in \{0,1\}$ 。} \\
&\cdots \quad \text{同算法 9-3} \quad \cdots \\
&\qquad \qquad \text{2.3.2（扰动动作）为目标动作 $\pi(S';\theta_{目标})$ 加受限的扰动，得到动作 $A'$ ，$(S,A,R,S')\in \mathcal B$ ；} \\
&\qquad \qquad \text{2.3.3（估计回报）$U \leftarrow R + \gamma \min_{i=0,1} q\left(S',A';\bold w^{(i)}\right)$ ，$(S,A,R,S')\in \mathcal B$ ；} \\
&\qquad \qquad \text{2.3.4（更新价值）更新 $\bold w^{(i)}$ 以减小 $\frac{1}{|\mathcal B|} \sum_{(S,A,R,S')\in \mathcal B} \left[U - q\left(S,A;\bold w^{(i)}\right)\right]^2$ ，$(i = 0,1)$ ；} \\
&\qquad \qquad \text{2.3.5（策略改进）更新 $\theta$ 以减小 $-\frac{1}{|\mathcal B|} \sum_{(S,A,R,S')\in \mathcal B} q\left(S,\pi(S;\theta);\bold w^{(0)}\right)$ ，} \\
&\qquad \qquad \qquad \;\, \text{如 $\theta \leftarrow \theta + \alpha^{(\theta)} \frac{1}{|\mathcal B|} \sum_{(S,A,R,S')\in \mathcal B} \nabla\pi(S;\theta) \left[\nabla_a q(S,a;\bold w^{(0)})\right]_{a=\pi(S;\theta)}$ ；} \\
&\qquad \qquad \text{2.3.6（更新目标）在恰当的时机更新目标网络和目标策略，$\bold w_{目标}^{(i)} \leftarrow (1-\alpha_{目标}) \bold w_{目标}^{(i)}$} \\
&\qquad \qquad \qquad \;\, \text{$+ \alpha_{目标}\bold w^{(i)}（i=0,1）$ ，$\theta_{目标} \leftarrow (1-\alpha_{目标}) \theta_{目标} + \alpha_{目标}\theta$ 。} \\
\rule[-5pt]{10mm}{0.1em} &\rule[-5pt]{265mm}{0.1em}
\end{split}
\; \\ \; \\</script>
					</div></div><h3><a name="三案例倒立摆控制pendulum-v0）" class="md-header-anchor"></a><span>三、案例：倒立摆控制（Pendulum-v0）</span></h3><p><span>本节使用 Gym 库中的倒立摆控制问题（Pendulum-v0）。该问题的对象是一根一端固定的棍子，以固定点为原点，垂直向上为 </span><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.36ex" viewBox="0 -500.4 572 585.5" role="img" focusable="false" style="vertical-align: -0.198ex;"><defs><path stroke-width="0" id="E4939-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 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y="0"></use><use xlink:href="#E4944-MJMAIN-30" x="9269" y="0"></use><use xlink:href="#E4944-MJMAIN-5D" x="9769" y="0"></use></g></svg></span><script type="math/tex">R_{t+1} \in [-\pi^2-6.404, 0]</script><span> ；任务是在给定的时间内（200 步）总收益越大越好，即相当于尽可能的保持木棍静止直立。其他更详细的数据可参阅</span><a href='https://github.com/openai/gym/blob/master/gym/envs/classic_control/pendulum.py' target='_blank' title='Pendulum-v0'><span>源代码</span></a><span>。</span></p><p><span>该问题的状态空间、动作空间、奖励空间都是连续的空间，问题整体的空间变得相当大；而且最大的力矩也无法将倒立摆从单边直接推至直立状态，因此智能体需要学会利用重力将倒立摆荡上至直立位置。</span></p><p><span>代码中类 </span><code>OrnsteinUhlenbeckProcess</code><span> 实现了 Ornstein Uhlenbeck 过程；智能体类 </span><code>DDPG</code><span> 实现了深度确定性策略梯度智能体类，以及智能体类 </span><code>TD3</code><span> 实现了双重延迟深度确定性策略梯度智能体类；前两者代码与书中基本一致，后者稍作修改，减少了一些代码量，但实现逻辑不变，此处不再展示。</span></p></div>
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